Diamond and $\beta$-tin structures of Si studied with quantum Monte Carlo calculations

We have used diffusion quantum Monte Carlo (DMC) calculations to study the pressure-induced phase transition from the diamond to $\beta$-tin structure in silicon. The calculations employ the pseudopotential technique and systematically improvable B-spline basis sets. We show that in order to achieve a precision of $1\mathrm{GPa}$ in the transition pressure the noncanceling errors in the energies of the two structures must be reduced to $30\mathrm{meV}∕\text{atom}$. Extensive tests on system size errors, nonlocal pseudopotential errors, basis-set incompleteness errors, and other sources of error, performed on periodically repeated systems of up to 432 atoms, show that all these errors together can be reduced to well below $30\mathrm{meV}∕\text{atom}$. The calculated DMC transition pressure is about $3–4\mathrm{GPa}$ higher than the accepted experimental range of values, and we argue that the discrepancy may be due to the fixed-node error inherent in DMC techniques.

Figure 1

Lower panel: DMC local energies as function of time (time $\text{step}=0.03\mathrm{a.u.}$) for the $\beta$-tin structure with $V=15{\mathrm{\AA }}^3∕\text{atom}$ (dotted line) and the diamond structure with $V=20{\mathrm{\AA }}^3∕\text{atom}$ (continuous line). Upper panel: the population of walkers for the $\beta$-tin structure (dotted line) and the diamond structure (continuous line).

Figure 2

DMC total energy per atom as a function of time step, with error bars showing the statistical errors. The calculations were performed using a cell containing 16 atoms in the $\beta$-tin structure at the volume $V=15{\mathrm{\AA }}^3∕\text{atom}$.

Figure 3

DMC energies for the $\beta$-tin structure at the volume $V=15{\mathrm{\AA }}^3∕\text{atom}$ as a function of the $c∕a$ ratio, performed with cells containing 128 atoms (●). DFT results performed with the equivalent mesh of $\mathbf{k}$-points are also shown (엯). DMC $\mathbf{k}$-points corrected results are shown as filled squares, and DFT results fully converged with respect to $\mathbf{k}$-point sampling as open squares. The lines are guides to the eye.

Figure 4

DMC total energies for the $\beta$-tin (∎) and the diamond (●) structures. The size of the points corresponds to about two standard deviations. The dashed and continuous lines are Birch-Murnaghan EOS curves fitted to the data.